History[edit]Archimedes used the me

History[edit]

Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.
Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[3] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy.[4] Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.[5] The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.[6] In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle.[7] Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century.[8] The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century.[9]

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent.[10] Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.

The modern foundations of mathematical analysis were established in 17th century Europe.[3] Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

In the 18th century, Euler introduced the notion of mathematical function.[11] Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816,[12] but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis.

In the middle of the 19th century Riemann introduced his theory of integration. The last third of the century saw the arithmetization of analysis by Weierstrass, who thought that geometric reasoning was inherently misleading, and introduced the "epsilon-delta" definition of limit. Then, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.

Also, "monsters" (nowhere continuous functions, continuous but nowhere differentiable functions, space-filling curves) began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue solved the problem of measure, and Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Important concepts[edit]
Metric spaces[edit]
Main article: Metric space
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).

Formally, A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a function

d colon M imes M
ightarrow mathbb{R}
such that for any x, y, z in M, the following holds:

d(x,y) = 0, if and only if x = y, (identity of indiscernibles),
d(x,y) = d(y,x), (symmetry) and
d(x,z) le d(x,y) + d(y,z) (triangle inequality) .
By taking the third property and letting z=x, it can be shown that d(x,y) ge 0 (non-negative).

Sequences and limits[edit]
Main article: Sequence
A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.

One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted

lim_{n oinfty} a_n = x.
Main branches[edit]
Real analysis[edit]
Main article: Real analysis
Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.[13][14] In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

Complex analysis[edit]
Main article: Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.[15] It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.

Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

Functional analysis[edit]
Main article: Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.[16][17] The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Differential equations[edit]
Main article: Differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.[18][19][20] Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation

Kết quả (Việt) 1:

History[edit]Archimedes used the method of exhaustion to compute the area inside a circle by finding the area of regular polygons with more and more sides. This was an early but informal example of a limit, one of the most basic concepts in mathematical analysis.Mathematical analysis formally developed in the 17th century during the Scientific Revolution,[3] but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy.[4] Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.[5] The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century.[6] In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century AD to find the area of a circle.[7] Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century.[8] The Indian mathematician Bhāskara II gave examples of the derivative and used what is now known as Rolle's theorem in the 12th century.[9]In the 14th century, Madhava of Sangamagrama developed infinite series expansions, like the power series and the Taylor series, of functions such as sine, cosine, tangent and arctangent.[10] Alongside his development of the Taylor series of the trigonometric functions, he also estimated the magnitude of the error terms created by truncating these series and gave a rational approximation of an infinite series. His followers at the Kerala school of astronomy and mathematics further expanded his works, up to the 16th century.The modern foundations of mathematical analysis were established in 17th century Europe.[3] Descartes and Fermat independently developed analytic geometry, and a few decades later Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.Trong thế kỷ 18, Euler giới thiệu khái niệm của toán học chức năng. [11] thực sự phân tích bắt đầu nổi lên như là một chủ đề độc lập khi Bernard Bolzano đã giới thiệu định nghĩa hiện đại của liên tục năm 1816, [12] nhưng công việc của Bolzano đã không trở thành rộng rãi được biết đến cho đến thập niên 1870. Năm 1821, Cauchy bắt đầu đưa tính toán trên một nền tảng logic vững chắc từ chối các nguyên tắc của quát của đại số sử dụng rộng rãi trong công việc trước đó, đặc biệt là bởi Euler. Thay vào đó, Cauchy công thức tính toán về ý tưởng hình học và Archimedes. Vì vậy, ông định nghĩa của liên tục yêu cầu một sự thay đổi infinitesimal trong x để tương ứng với một sự thay đổi infinitesimal trong y. Ông cũng giới thiệu khái niệm của dãy Cauchy, và bắt đầu chính thức lý thuyết giải tích phức. Poisson, Liouville, Fourier và những người khác nghiên cứu phương trình vi phân riêng phần và phân tích dao. Sự đóng góp của các nhà toán học và những người khác, chẳng hạn như Weierstrass, phát triển (ε, δ)-định nghĩa của phương pháp tiếp cận giới hạn, do đó thành lập trường hiện đại của giải tích toán học.Ở giữa thế kỷ 19 Riemann giới thiệu lý thuyết của ông của hội nhập. Thứ ba cuối của thế kỷ thấy arithmetization phân tích của Weierstrass, những người nghĩ rằng hình học lý luận vốn gây hiểu lầm, và giới thiệu định nghĩa "epsilon tam giác" giới hạn. Sau đó, nhà toán học bắt đầu lo lắng rằng họ đã giả định sự tồn tại của một liên tục của các số thực mà không có bằng chứng. Dedekind sau đó xây dựng số thực bởi Dedekind cắt giảm, trong đó vô tỉ số chính thức được xác định, mà phục vụ để điền vào các lỗ hổng"" giữa số hữu tỉ, do đó việc tạo ra một bộ hoàn chỉnh: liên tục của các số thực, mà đã được phát triển bởi Simon Stevin trong điều khoản của thập phân mở rộng. Khoảng thời gian đó, những nỗ lực để tinh chỉnh các định lý tích hợp Riemann đã dẫn đến nghiên cứu "kích thước" của tập hợp các discontinuities chức năng thực sự.Ngoài ra, "quái vật" (chức năng hư không liên tục, liên tục nhưng không có nơi nào khả vi chức năng, điền vào không gian đường cong) bắt đầu được điều tra. Trong bối cảnh này, Jordan phát triển lý thuyết đo, Cantor phát triển những gì bây giờ được gọi là lý thuyết tập hợp ngây thơ và Baire đã chứng minh định lý thể loại Baire. Đầu thế kỷ 20, tính toán được chính thức hóa bằng cách sử dụng một lý thuyết tập hợp tiên đề. Lebesgue giải quyết vấn đề của biện pháp, và giới thiệu Hilbert không gian Hilbert để giải quyết các phương trình tích phân. Ý tưởng của kì không gian vectơ là trong không khí, và trong thập niên 1920 Banach tạo ra giải tích hàm.Important concepts[edit]Metric spaces[edit]Main article: Metric spaceIn mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).Formally, A metric space is an ordered pair (M,d) where M is a set and d is a metric on M, i.e., a functiond colon M imes M
ightarrow mathbb{R}such that for any x, y, z in M, the following holds:d(x,y) = 0, if and only if x = y, (identity of indiscernibles),d(x,y) = d(y,x), (symmetry) andd(x,z) le d(x,y) + d(y,z) (triangle inequality) .By taking the third property and letting z=x, it can be shown that d(x,y) ge 0 (non-negative).Sequences and limits[edit]Main article: SequenceA sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denotedlim_{n oinfty} a_n = x.Main branches[edit]Real analysis[edit]Main article: Real analysisReal analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable.[13][14] In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.Complex analysis[edit]Main article: Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.[15] It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.
Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

Functional analysis[edit]
Main article: Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense.[16][17] The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Differential equations[edit]
Main article: Differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders.[18][19][20] Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation